%I
%S 1,2,3,2,4,3,5,3,6,5,4,7,5,4,8,6,5,9,7,5,10,8,7,6,11,8,7,6,12,10,9,7,
%T 13,10,9,7,14,11,9,8,15,12,11,10,8,16,13,12,11,9,17,13,12,11,9,18,15,
%U 13,12,10,19,15,13,12,10,20,16,15,13,11,21,17,16,15,14,11
%N Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176.
%C The dot product of the nth row of this triangle and the nth row of triangle A259176 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value is the sum of the rectangles between the xaxis and the horizontal legs of the symmetric representation of sigma(n). This is the companion computation to A283367.
%F T(n,k) = n  sum_{i=1..k1} f(n, 2*i) where f is defined in A237593.
%F A024916(n) = sum_{i=1..row(n)} (T(n,i))*S(n,i)) where S(n,i) refers to the triangle of A259176 and row(n) = floor(sqrt(8*n+1)1)/2).
%e The first horizontal leg of the symmetric representation of sigma(15) is at ycoordinate 15 and has length 8, and row 15 has 5 entries so that T(15,1) = 15 and T(15,5) = 8.
%e The first 16 rows of the irregular triangle:
%e 1
%e 2
%e 3 2
%e 4 3
%e 5 3
%e 6 5 4
%e 7 5 4
%e 8 6 5
%e 9 7 5
%e 10 8 7 6
%e 11 8 7 6
%e 12 10 9 7
%e 13 10 9 7
%e 14 11 9 8
%e 15 12 11 10 8
%e 16 13 12 11 9
%t (* function f[n,k] and its support functions are defined in A237593 *)
%t a283368[n_, k_] := n  Sum[f[n, 2i], {i, k1}]
%t TableForm[Table[a283368[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
%t Flatten[Table[a283368[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)
%Y Cf. A024916, A237593, A259176, A259177, A283367.
%K nonn,tabf
%O 1,2
%A _Hartmut F. W. Hoft_, Mar 06 2017
